Microcanonical Entropy — Boltzmann's S = k ln Ω

N harmonic oscillators at fixed energy E — density of states Ω(E,N) and entropy

6
10
Microstates Ω
-
Entropy S/k_B
-
Temperature k_BT/ε
-
Heat capacity C/Nk_B
-
Equipartition ⟨E⟩/N
-
ln Ω = S/k_B
-
The microcanonical ensemble fixes total energy E, volume V, and particle number N. For N harmonic oscillators with energy E (in units of ℏω), the number of microstates is Ω(E,N) = C(E+N-1, N-1) — distributing E indistinguishable quanta among N oscillators. Boltzmann entropy S = kB ln Ω. Temperature emerges as 1/T = ∂S/∂E|N,V, giving kBT = ε/ln(1+N/E) for Einstein solid. As E→∞, T→∞ and C→NkB (Dulong-Petit law). Left: microstate distribution (bar chart). Right: selected curve vs E.